Problem 30 - College Algebra

Question

For the following exercises, expand the binomial. $$(9 b+1)^{2}$$

Step-by-Step Solution

Answer

The expanded form is \(81b^2 + 18b + 1\).

Step 1: Identify the Binomial

The given binomial is \((9b + 1)\). We are tasked with expanding \((9b + 1)^2\).

Step 2: Use the Binomial Expansion Formula

Recall the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 9b\) and \(b = 1\).

Step 3: Calculate Each Term

- First term: \(a^2 = (9b)^2 = 81b^2\).- Second term: \(2ab = 2 \times 9b \times 1 = 18b\).- Third term: \(b^2 = 1^2 = 1\).

Step 4: Combine the Terms

Combine all the terms from Step 3 to get the expanded form: \(81b^2 + 18b + 1\).

Key Concepts

Understanding Algebra in Binomial Expansion

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is the underlying structure of mathematics when we need to solve problems involving unknown variables. In binomial expansion, particularly, algebra plays a vital role as it helps to manage and manipulate expressions involving two terms or variables.

• Binomials are expressions containing two terms. We often write them in the form \(a + b\) or \(a - b\).
• Algebraic manipulation allows us to expand a binomial expression, such as \(9b + 1\), using specific formulas like the square of a binomial.
• In the given example, algebraic principles are applied systematically to identify, rewrite, and expand the given binomial expression \(9b + 1\), stepping through each component logically until reaching a final expression.

Albeit foundational, these principles are incredibly powerful in simplifying and solving complex mathematical problems by systematically transforming and combining terms.

Understanding Polynomials in Expansion

Polynomials are mathematical expressions consisting of variables, coefficients, and exponents that can appear in various powers. When binomials are expanded, they result in polynomials. The task of expanding \(9b + 1\) involves determining its polynomial representation, which involves breaking it down to simpler terms and then combining them.

• In the original expression \(9b + 1\), each term can be considered part of a polynomial once expanded.
• The expansion results in the polynomial \(81b^2 + 18b + 1\), where each element is a term of the polynomial.
• Polynomials are often arranged in decreasing powers of the variable, which makes it easier to follow and apply algebraic operations consistently.

Understanding polynomials is essential because it helps in visualizing how individual algebraic terms come together as a whole, providing a broader view of the mathematical expression at play.

Role of Exponents in Binomial Expansions

Exponents are fundamental in algebra, especially in expressions where terms are multiplied by themselves. An exponent indicates how many times to use the base in a multiplication. In our context, expanding \(9b + 1\) involved using exponents to simplify and calculate each term of the polynomial.

• The binomial \(9b + 1\) squared, or raised to the power of two, means multiplying the entire expression by itself.
• Applying the exponent, \( (9b)^2\) expands to \81b^2\ where \((9b)\) is the base, and \(2\) is the exponent indicating squaring.
• A thorough understanding of how exponents interact with coefficients and variables is critical for correctly applying them in the context of polynomial expansions.

Exponential rules allow us to handle potentially complex computations with simplicity, converting a seemingly large process into manageable steps for solving.